We recall the definitions of index and vorticity of a vector field
(see also [1]}).
To every isolated zero \(p\in M\)  of a smooth vector field \(u\)  we can associate an integer number, called the index of \(u\)  at \(p\) , in the following way.
Let \(\delta > 0\)  be a small parameter, such that
the ball \(B_\delta (p)\)  does not contain any zero of \(u\) 
other than \(p\) . Using normal geodesic coordinates,
we may represent \(u\)  by a map \(v\colon B_\delta (0)\subseteq \mathbb {R}^2\rightarrow \mathbb {R}^2\) ,
as in (REF ). We have \(\left| v \right| = \left| u \right|_g > 0\)  on \(\partial V\) 
and hence, the map \(v/\left| v \right|\colon \partial _\delta (0)\rightarrow \mathbb {S}^1\) 
is well-defined and smooth. We define the index of \(u\)  at \(p\) 
as the topological degree of \(v/\left| v \right|\)  on \(\partial B_\delta (p)\) ,
\(\operatorname{ind}(u, \, \partial B_\delta (p)):= \deg \left(\frac{v}{\left| v \right|}, \, \partial B_\delta (p), \, \mathbb {S}^1\right)\) 
